Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

Consistent mean field optimality conditions for interacting agent systems

Pages: 1095 – 1108



Michael Herty (Department of Mathematics, RWTH Aachen University, Aachen, Germany)

Christian Ringhofer (Department of Mathematics, Arizona State University, Tempe, Az., U.S.A.)


We are interested in the derivation of optimality conditions for controlled interacting agent systems. We establish the relation between mean field optimality conditions and the optimality condition of the mean field control problem. This link is important for many recently published articles on control strategies for agent based systems since it establishes the precise relation between multipliers for the individual agents and the probability density distribution of the multipliers in the mean field limit. The relation to different notions of differentiability are also shown.


kinetic equations, particle systems, optimal control, meanfield limits

2010 Mathematics Subject Classification

49J15, 49J20, 82B40

This work has been supported by NSF RNMS grant No. 1107291 (KI-Net) and DFG HE5386/14-15, BMBF 05M18PAA, DFG-GRK 2326 and ID390621612 Cluster of Excellence Internet of Production (IoP).

Received 29 November 2018

Accepted 1 May 2019

Published 25 October 2019